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The twisted cubic in PG(3,q) and translation spreads in H(q)

2005
*
Discrete Mathematics
*

Using

doi:10.1016/j.disc.2005.03.010
fatcat:3a4xumcd6bfctdv4sol43bq2ci
*the*connection between*translation**spreads*of*the*classical generalized hexagon*H*(*q*)*and**the**twisted**cubic*of*PG*(*3*,*q*), established*in*[European J. ... Combin. 23 (2002) 367-376], we prove that if*q*n ≡ 1 (mod*3*),*q*odd,*q*4n 2 − 8n + 2*and*n > 2, then*H*(*q*n ) does not admit an F*q*-*translation**spread*. ...*3*,*q*n ) of rank 2n whose points belong to imaginary chords of a*twisted**cubic*C of*PG*(*3*,*q*n ),*and*conversely. ...##
###
On the Twisted Cubic of PG(3, q)

2003
*
Journal of Algebraic Combinatorics
*

*In*this paper we classify

*the*lines of

*PG*(

*3*,

*q*) whose points belong to imaginary chords of

*the*

*twisted*

*cubic*of

*PG*(

*3*,

*q*). ... Relying on this classification result, we obtain a complete classification of semiclassical

*spreads*of

*the*generalized hexagon

*H*(

*q*). ... Let¯ be

*the*

*twisted*

*cubic*of

*PG*(

*3*, F) defined by , where F is

*the*algebraic closure of GF(

*q*). ...

##
###
An investigation of the tangent splash of a subplane of $$\mathrm{PG}(2,q^3)$$ PG ( 2 , q 3 )

2014
*
Designs, Codes and Cryptography
*

2,

doi:10.1007/s10623-014-9971-3
fatcat:4yosskmsavhrxl3rvotiucbhgq
*q**3*)*in**PG*(6,*q*). ...*In**PG*(2,*q**3*), let π be a subplane of order*q*that is tangent to ∞ .*The*tangent splash of π is defined to be*the*set of*q*2 + 1 points on ∞ that lie on a line of π. ... One for pointing out*the*important relationship between tangent splashes*and*linear sets,*and*that a number of our results could be proved more directly using*the*theory of linear sets. ...##
###
Absolute points of correlations of $$PG(3,q^n)$$

2020
*
Journal of Algebraic Combinatorics
*

As an application we show that, for

doi:10.1007/s10801-020-00970-3
fatcat:gtz2spm5r5cz3o2e566prrimou
*q*even, some of these sets are related to*the*Segre's (2*h*+ 1)-arc of*PG*(*3*, 2 n )*and*to*the*Lüneburg*spread*of*PG*(*3*, 2 2h+1 ). ...*In*this paper, we completely determine*the*sets of*the*absolute points of degenerate correlations, different from degenerate polarities, of a projective space*PG*(*3*,*q*n ). ... Degenerate -quadrics*and*Lüneburg*spread*of*PG*(*3*, 2 n ).*In*1965,*H*. ...##
###
An investigation of the tangent splash of a subplane of PG(2,q^3)
[article]

2014
*
arXiv
*
pre-print

2,

arXiv:1303.5509v2
fatcat:mluj5kn6e5he3b2524356ob444
*q*^*3*)*in**PG*(6,*q*). ...*In**PG*(2,*q*^*3*), let π be a subplane of order*q*that is tangent to ℓ_infty.*The*tangent splash of π is defined to be*the*set of*q*^2+1 points on ℓ_infty that lie on a line of π. ... One for pointing out*the*important relationship between tangent splashes*and*linear sets,*and*that a number of our results could be proved more directly using*the*theory of linear sets. ...##
###
Ruled cubic surfaces in PG(4,q), Baer subplanes of PG(2,q2) and Hermitian curves

2002
*
Discrete Mathematics
*

*The*Andrà e=Bruck

*and*Bose representation of

*PG*(2;

*q*2 ) involves a regular

*spread*

*in*

*PG*(

*3*;

*q*). ... (Utilitas Math. 17 (1980) 65) that non-a ne Baer subplanes of

*PG*(2;

*q*2 ) are represented by certain ruled

*cubic*surfaces

*in*

*the*Andrà e=Bruck

*and*Bose representation of

*PG*(2;

*q*2 )

*in*

*PG*(4;

*q*) (Math. ... Hence

*H*∈ PGL(2;

*q*), so that

*the*ruled

*cubic*surface so determined is a V

*3*2 contained

*in*

*PG*(4;

*q*). ...

##
###
Fq-pseudoreguli of PG(3,q3) and scattered semifields of order q6

2011
*
Finite Fields and Their Applications
*

*The*known examples of such semifields are some Knuth semifields, some Generalized

*Twisted*Fields

*and*

*the*semifields recently constructed

*in*Marino et al. (

*in*press) [12] for

*q*≡ 1 (mod

*3*). ... Here, we construct new infinite families of rank two scattered semifields for any

*q*odd prime power, with

*q*≡ 1 (mod

*3*); for any

*q*= 2 2h , such that

*h*≡ 1 (mod

*3*)

*and*for any

*q*=

*3*

*h*with

*h*≡ 0 (mod

*3*... A way to construct scattered F

*q*-linear sets of rank 6

*in*

*PG*(

*3*,

*q*

*3*) is

*the*following. Let r

*and*r be two disjoint lines of

*PG*(

*3*,

*q*

*3*) =

*PG*(V ), say r =

*PG*(U )

*and*r =

*PG*(U ), with V = U ⊕ U . ...

##
###
The tangent splash in (6,q)
[article]

2013
*
arXiv
*
pre-print

*In*

*the*Bruck-Bose representation of

*PG*(2,

*q*^

*3*)

*in*

*PG*(6,

*q*), we investigate

*the*interaction between

*the*ruled surface corresponding to B

*and*

*the*planes corresponding to

*the*tangent splash of B. ... Let B be a subplane of

*PG*(2,

*q*^

*3*) of order

*q*that is tangent to ℓ_∞. Then

*the*tangent splash of B is defined to be

*the*set of

*q*^2+1 points of ℓ_∞ that lie on a line of B. ... If

*q*is even, then

*the*tangents to a

*twisted*

*cubic*lie

*in*a regulus [9,

*q*

*3*) ,

*H** contains

*the*

*twisted*

*cubic*N * ,

*and*so

*H** contains

*the*transversal points P, P

*q*, P

*q*2 . ...

##
###
The exterior splash in PG(6,q): Special conics
[article]

2014
*
arXiv
*
pre-print

Exterior splashes are projectively equivalent to scattered linear sets of rank

arXiv:1410.4269v1
fatcat:exrug5lff5hptna72c4aeaww6e
*3*, covers of*the*circle geometry CG(*3*,*q*),*and*hyper-reguli of*PG*(5,*q*). ...*In*this article we use*the*Bruck-Bose representation*in**PG*(6,*q*) to give a geometric characterisation of special conics of π*in*terms of*the*covers of*the*exterior splash of π. ...*In**PG*(6,*q*), [C i ] is an X-special*twisted**cubic**in*a*3*-space Π i about a plane of X. By Theorem 3.10,*the*line P i L ii is*in**the**3*-space Π i , hence Π i = Σ ii . ...##
###
The exterior splash in PG(6,q): Transversals
[article]

2014
*
arXiv
*
pre-print

*In*

*PG*(6,

*q*), an exterior splash S has two sets of cover planes (which are hyper-reguli)

*and*we show that each set has three unique transversals lines

*in*

*the*

*cubic*extension

*PG*(6,

*q*^

*3*). ... Exterior splashes are projectively equivalent to scattered linear sets of rank

*3*, covers of

*the*circle geometry CG(

*3*,

*q*),

*and*hyper-reguli

*in*

*PG*(5,

*q*). ... An S-special

*twisted*

*cubic*is a

*twisted*

*cubic*N

*in*a

*3*-space of

*PG*(6,

*q*)\Σ ∞ about a plane of S, such that

*the*extension of N to

*PG*(6,

*q*

*3*) meets

*the*transversals of S. ...

##
###
On symplectic semifield spreads of PG(5,q 2), q odd

2018
*
Forum mathematicum
*

We prove that there exist exactly three non-equivalent symplectic semifield

doi:10.1515/forum-2016-0133
fatcat:2fg7js2465eptofbpkkcec7qta
*spreads*of {\operatorname{*PG*}(5,*q*^{2})} , for {*q*^{2}>2\cdot*3*^{8}} odd, whose associated semifield has center containing {\mathbb ... Equivalently, we classify, up to isotopy, commutative semifields of order {*q*^{6}} , for {*q*^{2}>2\cdot*3*^{8}} odd, with middle nucleus containing {\mathbb{F}_{*q*^{2}}}*and*center containing {\mathbb{F}_{ ... associated with a*twisted*field A*and*a*spread*comprising*the*constructions BH, LMPT , ZP. ...##
###
On projective q^r-divisible codes
[article]

2019
*
arXiv
*
pre-print

One example are upper bounds on

arXiv:1912.10147v1
fatcat:fpig43peijgvrn5dngi5aaa6q4
*the*cardinality of partial*spreads*. Here we survey*the*known results on*the*possible lengths of projective*q*^r-divisible linear codes. ... Especially,*q*^r-divisible projective linear codes, where r is some integer, arise*in*many applications of collections of subspaces*in*F_q^v. ... x) = f ,*and*(*3*) min{ x∈*H*w(x) |*H*∈*H*} = m, where*H*is*the*set of hyperplanes of*PG*(v − 1,*q*) . ...##
###
Page 2702 of Mathematical Reviews Vol. , Issue 2003d
[page]

2003
*
Mathematical Reviews
*

By representing

*H*(*q*) as a coset geometry, we obtain a characterization of a*translation**spread**in*terms of a set of points of*PG*(*3*,*q*) which belong to imaginary chords of a*twisted**cubic**and*we construct ... Let 2, =*PG*(*3*,*q*) be*the*hyperplane at infinity,*and*let L be*the*fixed regular*spread*of x. used*in**the*above representation. ...##
###
Page 10043 of Mathematical Reviews Vol. , Issue 2004m
[page]

2004
*
Mathematical Reviews
*

(I-NAPL2; Caserta) On

*the**twisted**cubic*of*PG*(*3*,*q*¢). (English summary) J. Algebraic Combin. 18 (2003), no.*3*, 255-262. ... Lines of*PG*(*3*,*q*) whose points belong to imaginary chords of*the**twisted**cubic*£ of*PG*(*3*,*q*) are classified by proving that if / is a line whose points belong to imaginary chords of £, then either / is an ...##
###
Linear sets in finite projective spaces

2010
*
Discrete Mathematics
*

Also, some applications of

doi:10.1016/j.disc.2009.04.007
fatcat:tuvgbx5qtbgkxmn74ywas2aysm
*the*theory of linear sets are investigated: blocking sets*in*Desarguesian planes, maximum scattered linear sets,*translation*ovoids of*the*Cayley Hexagon,*translation*ovoids ...*In*this paper linear sets of finite projective spaces are studied*and**the*"dual" of a linear set is introduced. ... Acknowledgement This work was supported by*the*Research Project of MIUR (Italian Office for University*and*Research) ''Strutture geometriche, combinatoria e loro applicazioni''. ...
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